Subgroups — C₄₇×#18402 #26058100

Order 12032.

1174 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3497	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#3206	C₂	1	2	yes	maximal, complemented
3	C₄₇×#2567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#1604	C₂	1	8	yes	maximal, complemented
5	C₄₇×#1502	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×2₋¹⁺⁴×C₄₇	—	2	1	no	—
8	C₄₇×#550	—	2	2	no	—
9	D₄×Q₈×C₄₇	—	2	2	no	—
10	C₄₇×#548	—	2	4	no	—
11	C₄₇×#537	—	2	1	no	—
12	C₄₇×#537	C₂²	1	4	yes	—
13	C₄₇×#532	—	2	2	no	—
14	C₄₇×#532	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#492	—	2	2	no	—
18	C₄₇×#487	—	2	2	no	—
19	C₄₇×#461	—	2	4	no	—
20	C₄₇×#448	C₂²	1	8	yes	—
21	C₄₇×#431	C₂²	1	2	yes	—
22	C₄₇×#395	C₂²	1	4	yes	—
23	C₄₇×#391	—	2	1	no	—
24	C₄₇×#389	—	2	2	no	—
25	C₄₇×#384	—	2	1	no	—
26	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#327	C₂²	1	8	yes	—
28	C₄₇×#324	C₂²	1	4	yes	—
29	2₋¹⁺⁴×C₄₇	—	4	4	no	—
30	2₋¹⁺⁴×C₄₇	—	4	4	no	—
31	C₂×#41×C₄₇	—	2	4	no	—
32	C₂×#41×C₄₇	—	2	4	no	—
33	C₂×#41×C₄₇	—	2	2	no	—
34	C₂²×Q₈×C₄₇	—	2	4	no	—
35	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
37	C₂×SD₁₆×C₄₇	—	2	8	no	—
38	C₄⋊Q₈×C₄₇	—	2	2	no	—
39	C₄⋊Q₈×C₄₇	—	2	2	no	—
40	C₄⋊Q₈×C₄₇	—	2	2	no	—
41	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
42	#127×C₄₇	—	2	2	no	—
43	#127×C₄₇	D₄	1	2	yes	—
44	#124×C₄₇	—	2	4	no	—
45	#124×C₄₇	—	2	2	no	—
46	C₂²⋊Q₈×C₄₇	—	2	4	no	—
47	C₂²⋊Q₈×C₄₇	—	2	8	no	—
48	#121×C₄₇	—	2	4	no	—
49	#121×C₄₇	—	2	8	no	—
50	#121×C₄₇	—	2	2	no	—
51	#121×C₄₇	—	2	4	no	—
52	#121×C₄₇	C₂³	1	4	yes	—
53	C₄×Q₈×C₄₇	—	2	2	no	—
54	C₄×Q₈×C₄₇	—	2	2	no	—
55	C₄×D₄×C₄₇	—	2	4	no	—
56	C₄×D₄×C₄₇	—	2	4	no	—
57	C₄×D₄×C₄₇	—	2	2	no	—
58	C₄×D₄×C₄₇	D₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
65	C₂×C₄²×C₄₇	—	2	1	no	—
66	C₂×C₄²×C₄₇	D₄	1	2	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₄₇	—	2	2	no	—
69	#103×C₄₇	—	2	8	no	—
70	C₂²⋊C₈×C₄₇	C₂³	1	4	yes	—
71	C₄×C₈×C₄₇	—	2	2	no	—
72	#95×C₄₇	—	2	4	no	—
73	#95×C₄₇	C₂³	1	2	yes	—
74	#41×C₄₇	—	4	8	no	—
75	#41×C₄₇	—	4	16	no	—
76	#41×C₄₇	—	4	8	no	—
77	#41×C₄₇	—	4	4	no	—
78	#41×C₄₇	—	4	4	no	—
79	C₂×Q₈×C₄₇	—	2	4	no	—
80	C₂×Q₈×C₄₇	—	2	2	no	—
81	C₂×Q₈×C₄₇	—	2	2	no	—
82	C₂×Q₈×C₄₇	—	2	2	no	—
83	C₂×Q₈×C₄₇	—	2	4	no	—
84	C₂×Q₈×C₄₇	—	4	16	no	—
85	C₂×Q₈×C₄₇	—	8	2	no	—
86	C₂×Q₈×C₄₇	SD₁₆	1	4	yes	complemented
87	C₂×D₄×C₄₇	—	2	4	no	—
88	C₂×D₄×C₄₇	—	2	4	no	—
89	C₂×D₄×C₄₇	—	2	4	no	—
90	C₂×D₄×C₄₇	—	2	2	no	—
91	C₂×D₄×C₄₇	—	2	2	no	—
92	C₂×D₄×C₄₇	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₄₇	—	2	8	no	—
94	C₂²×C₄×C₄₇	—	2	2	no	—
95	C₂²×C₄×C₄₇	—	2	4	no	—
96	C₂²×C₄×C₄₇	—	2	2	no	—
97	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
102	SD₁₆×C₄₇	—	8	8	no	—
103	C₂×C₈×C₄₇	—	2	4	no	—
104	C₄⋊C₄×C₄₇	—	2	4	no	—
105	C₄⋊C₄×C₄₇	—	2	4	no	—
106	C₄⋊C₄×C₄₇	—	2	2	no	—
107	C₄⋊C₄×C₄₇	—	2	2	no	—
108	C₄⋊C₄×C₄₇	—	2	4	no	—
109	C₄⋊C₄×C₄₇	—	4	1	no	—
110	C₄⋊C₄×C₄₇	—	4	2	no	—
111	C₄⋊C₄×C₄₇	—	4	1	no	—
112	C₄⋊C₄×C₄₇	—	8	4	no	—
113	C₄⋊C₄×C₄₇	C₂×D₄	1	4	yes	—
114	C₂²⋊C₄×C₄₇	—	2	8	no	—
115	C₂²⋊C₄×C₄₇	—	2	4	no	—
116	C₂²⋊C₄×C₄₇	—	2	4	no	—
117	C₄²×C₄₇	—	2	2	no	—
118	C₄²×C₄₇	—	2	2	no	—
119	C₄²×C₄₇	—	4	2	no	—
120	C₄²×C₄₇	C₂×D₄	1	2	yes	—
121	C₄²×C₄₇	C₂×D₄	1	2	yes	—
122	C₂³×C₄₇	—	2	4	no	—
123	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₄₇	—	2	8	no	—
125	Q₈×C₄₇	—	4	8	no	—
126	Q₈×C₄₇	—	4	8	no	—
127	Q₈×C₄₇	—	4	4	no	—
128	D₄×C₄₇	—	4	4	no	—
129	D₄×C₄₇	—	4	8	no	—
130	D₄×C₄₇	—	4	4	no	—
131	D₄×C₄₇	—	4	4	no	—
132	D₄×C₄₇	—	4	4	no	—
133	D₄×C₄₇	—	4	4	no	—
134	D₄×C₄₇	—	8	4	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	4	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	2	4	no	—
141	C₂×C₄×C₄₇	—	4	1	no	—
142	C₂×C₄×C₄₇	—	4	16	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	—	4	8	no	—
146	C₂×C₄×C₄₇	—	4	2	no	—
147	C₂×C₄×C₄₇	—	4	4	no	—
148	C₂×C₄×C₄₇	—	4	2	no	—
149	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
150	C₂×C₄×C₄₇	#120	1	2	yes	—
151	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
153	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
154	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
155	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
156	C₈×C₄₇	—	4	4	no	—
157	#18402	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	—	4	4	no	—
161	C₂²×C₄₇	—	4	4	no	—
162	C₂²×C₄₇	—	4	4	no	—
163	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
164	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
165	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₄×C₄₇	—	2	4	no	—
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	4	4	no	—
170	C₄×C₄₇	—	4	4	no	—
171	C₄×C₄₇	—	4	2	no	—
172	C₄×C₄₇	—	8	2	no	—
173	C₄×C₄₇	#450	1	4	yes	—
174	C₄×C₄₇	#456	1	2	yes	—
175	#3497	C₂×C₄₇	1	1	yes	characteristic
176	#3206	C₂×C₄₇	1	2	yes	complemented
177	#2567	C₂×C₄₇	1	1	yes	characteristic, complemented
178	#1604	C₂×C₄₇	1	8	yes	complemented
179	#1502	C₂×C₄₇	1	2	yes	complemented
180	#1461	C₂×C₄₇	1	1	yes	characteristic, complemented
181	C₂×C₄₇	—	2	1	no	—
182	C₂×C₄₇	—	2	1	no	—
183	C₂×C₄₇	—	8	4	no	—
184	C₂×C₄₇	C₂×#450	1	1	yes	characteristic
185	C₂×C₄₇	C₂×#456	1	1	yes	characteristic
186	C₂×C₄₇	#3027	1	1	yes	characteristic
187	C₂×2₋¹⁺⁴	—	2	1	no	—
188	#550	—	2	2	no	—
189	D₄×Q₈	—	2	2	no	—
190	#548	—	2	4	no	—
191	#537	—	2	1	no	—
192	#537	C₂²×C₄₇	1	4	yes	—
193	#532	—	2	2	no	—
194	#532	C₂²×C₄₇	1	2	yes	—
195	C₂×C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
196	C₂×C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
197	#492	—	2	2	no	—
198	#487	—	2	2	no	—
199	#461	—	2	4	no	—
200	#448	C₂²×C₄₇	1	8	yes	—
201	#431	C₂²×C₄₇	1	2	yes	—
202	#395	C₂²×C₄₇	1	4	yes	—
203	#391	—	2	1	no	—
204	#389	—	2	2	no	—
205	#384	—	2	1	no	—
206	C₄×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
207	#327	C₂²×C₄₇	1	8	yes	—
208	#324	C₂²×C₄₇	1	4	yes	—
209	C₄₇	#18402	1	1	yes	characteristic, direct product factor, Sylow
210	2₋¹⁺⁴	—	4	4	no	—
211	2₋¹⁺⁴	—	4	4	no	—
212	C₂×#41	—	2	4	no	—
213	C₂×#41	—	2	4	no	—
214	C₂×#41	—	2	2	no	—
215	C₂²×Q₈	—	2	4	no	—
216	C₂²×Q₈	D₄×C₄₇	1	1	yes	characteristic
217	C₂²×Q₈	C₂³×C₄₇	1	2	yes	—
218	C₂×SD₁₆	—	2	8	no	—
219	C₄⋊Q₈	—	2	2	no	—
220	C₄⋊Q₈	—	2	2	no	—
221	C₄⋊Q₈	—	2	2	no	—
222	C₄⋊Q₈	D₄×C₄₇	1	2	yes	—
223	#127	—	2	2	no	—
224	#127	D₄×C₄₇	1	2	yes	—
225	#124	—	2	4	no	—
226	#124	—	2	2	no	—
227	C₂²⋊Q₈	—	2	4	no	—
228	C₂²⋊Q₈	—	2	8	no	—
229	#121	—	2	4	no	—
230	#121	—	2	8	no	—
231	#121	—	2	2	no	—
232	#121	—	2	4	no	—
233	#121	C₂³×C₄₇	1	4	yes	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×Q₈	—	2	2	no	—
236	C₄×D₄	—	2	4	no	—
237	C₄×D₄	—	2	4	no	—
238	C₄×D₄	—	2	2	no	—
239	C₄×D₄	D₄×C₄₇	1	4	yes	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄⋊C₄	—	2	4	no	—
244	C₂×C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
245	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
246	C₂×C₄²	—	2	1	no	—
247	C₂×C₄²	D₄×C₄₇	1	2	yes	—
248	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
249	C₄⋊C₈	—	2	2	no	—
250	#103	—	2	8	no	—
251	C₂²⋊C₈	C₂³×C₄₇	1	4	yes	—
252	C₄×C₈	—	2	2	no	—
253	#95	—	2	4	no	—
254	#95	C₂³×C₄₇	1	2	yes	—
255	#41	—	4	8	no	—
256	#41	—	4	16	no	—
257	#41	—	4	8	no	—
258	#41	—	4	4	no	—
259	#41	—	4	4	no	—
260	C₂×Q₈	—	2	4	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	4	no	—
265	C₂×Q₈	—	4	16	no	—
266	C₂×Q₈	—	8	2	no	—
267	C₂×Q₈	SD₁₆×C₄₇	1	4	yes	complemented
268	C₂×D₄	—	2	4	no	—
269	C₂×D₄	—	2	4	no	—
270	C₂×D₄	—	2	4	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	C₂×D₄×C₄₇	1	4	yes	—
274	C₂²×C₄	—	2	8	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
280	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
281	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
282	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
283	SD₁₆	—	8	8	no	—
284	C₂×C₈	—	2	4	no	—
285	C₄⋊C₄	—	2	4	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	4	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	8	4	no	—
294	C₄⋊C₄	C₂×D₄×C₄₇	1	4	yes	—
295	C₂²⋊C₄	—	2	8	no	—
296	C₂²⋊C₄	—	2	4	no	—
297	C₂²⋊C₄	—	2	4	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	4	2	no	—
301	C₄²	C₂×D₄×C₄₇	1	2	yes	—
302	C₄²	C₂×D₄×C₄₇	1	2	yes	—
303	C₂³	—	2	4	no	—
304	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
305	Q₈	—	2	8	no	—
306	Q₈	—	4	8	no	—
307	Q₈	—	4	8	no	—
308	Q₈	—	4	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	8	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	4	no	—
315	D₄	—	8	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	16	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	8	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
331	C₂×C₄	#120×C₄₇	1	2	yes	—
332	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
333	C₂×C₄	C₂×SD₁₆×C₄₇	1	4	yes	—
334	C₂×C₄	C₂×SD₁₆×C₄₇	1	2	yes	—
335	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
336	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
337	C₈	—	4	4	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	4	4	no	—
343	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
344	C₂²	C₂²×SD₁₆×C₄₇	1	1	yes	characteristic
345	C₂²	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
346	C₄	—	2	4	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	4	4	no	—
350	C₄	—	4	4	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	2	no	—
353	C₄	C₄₇×#450	1	4	yes	—
354	C₄	C₄₇×#456	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	8	4	no	—
358	C₂	C₂×C₄₇×#450	1	1	yes	characteristic
359	C₂	C₂×C₄₇×#456	1	1	yes	characteristic
360	C₂	C₄₇×#3027	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₄₇×#3497	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₄₇×#3206	C₂	semidirect
3. C₄₇×#2567	C₂	semidirect
4. C₄₇×#1604	C₂	semidirect
5. C₄₇×#1502	C₂	semidirect
6. C₄₇×#1461	C₂	semidirect
86. C₂×Q₈×C₄₇	SD₁₆	semidirect
157. #18402	C₄₇	direct (N × H)
176. #3206	C₂×C₄₇	semidirect
177. #2567	C₂×C₄₇	semidirect
178. #1604	C₂×C₄₇	semidirect
179. #1502	C₂×C₄₇	semidirect
180. #1461	C₂×C₄₇	semidirect
209. C₄₇	#18402	direct (N × H)
267. C₂×Q₈	SD₁₆×C₄₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄₇×#3497	2
C₄₇×#3206	2
C₄₇×#2567	2
C₄₇×#1604	2
C₄₇×#1502	2
C₄₇×#1461	2
#18402	47

Subgroups — C47×#18402 #26058100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#18402 #26058100